It is not true that dV in the volume integral ∫dV means d(xyz). Instead, it means ∫dxdydz: the infinitesimal volume dV is the same thing as the product of the three infinitesimal "linear factors": it makes absolutely no sense to go from the infinitesimal dV to the "whole" V and then "differentiate it back".
A triple integral is just a sequence of three integrations in a row. You may first integrate over z, then over y, then over x. Alternatively, you may often use more convenient coordinates – axial, spherical, or others – and make the calculation more tractable. Many of those triple integrals are exactly solvable, others are not. It's a purely mathematical question which of them may be expressed in terms of simple functions.
In these integrals, while calculating the moment of inertia, you may write down the general formula
I=∫dVρr2
where
ρ is a mass density at the given point (where the small volume
dV is located). If
ρ is equal to zero except for an interval, you may replace the integral above, which was assumed to be from
−∞ to
+∞ so that the whole space is covered, by the integral over the interval where
ρ is nonzero.
This post imported from StackExchange Mathematics at 2014-06-02 20:29 (UCT), posted by SE-user Luboš Motl